The model for a rarefied mono atomic gases under binary elastic interactions, originally introduced by L. Boltzmann in 1872 [1]}, and analytically studied by many authors including Carleman, Wild, Chapman, Cowling, Cercignani, Illner, Pulvirenti [2]}, [3]}, [4]}, [5]}, [6]} to name a few, is written as follows
\({\frac{\partial f}{\partial t} + v\cdot \nabla _x f= a^2 N \int _{{\mathbb {R}}^3\times \mathbb {S}^{2}_{+}}({}^{\prime }\!f{}^{\prime }\!f_* - ff_*) B(|u|, \hat{u}\cdot \eta ) \,dv_*d\eta }\) 
